Motivic homotopy theory for perfect schemes
Christian Dahlhausen, Jeroen Hekking, Storm Wolters
TL;DR
We construct a perfect version of Morel–Voevodsky motivic homotopy theory over a perfect base in positive characteristic by inverting universal homeomorphisms via perfection, yielding a stable category $\mathrm{S}\mathscr{H}^{\mathrm{pf}}(S)$ with a six-functor formalism. A key result is that after inverting $p$, the classical motivic homotopy category agrees with the perfect theory: $\mathrm{S}\mathscr{H}(S_0)[1/p] \simeq \mathrm{S}\mathscr{H}^{\mathrm{pf}}(S) \simeq \mathrm{S}\mathscr{H}_{\mathrm{idh}}(S_0)$. The paper develops the $\mathrm{pcdh}$ and $\mathrm{idh}$ topologies, proves hypercompleteness, and establishes adjunctions between the classical and perfect theories, ensuring a coherent coefficient-system framework. It also analyzes Thom spaces, perfection of closed immersions, and fibered/perfect motivic spaces to support stability and functoriality, and shows connective K-theory is representable in the unstable pf motivic category via $\mathbb{Z}\times\mathrm{BGL}$, while full $K$-theory is not representable in the pf setting. Overall, the results provide a robust, $p$-localized, perfect-geometry–oriented motivic framework with a concrete comparison to the classical theory and applications to K-theory in positive characteristic.
Abstract
We construct a perfect version of Morel--Voevodsky's motivic homotopy category over a perfect base scheme in positive characteristic. By checking the axioms of a coefficient system, we establish a six-functor formalism. We show that multiplication by $p$ is already invertible in the perfect motivic homotopy catgory. By work of Elmanto--Khan the functor sending an $\mathbb{F}_p$-scheme $S$ to the category $\mathrm{S}\mathcal{H}(S)[1/p]$ is invariant under universal homeomorphisms, hence under perfections. Our result gives an explicit model for the localization of $\mathrm{S}\mathcal{H}$ at the universal homeomorphisms, which we conclude is the same as $\mathrm{S}\mathcal{H}[1/p]$.